Method and system for calibrating a downhole imaging tool

ABSTRACT

System and methods of generating calibrated downhole images of a subterranean formation ( 110 ) surrounding a wellbore ( 105 ). The method involves measuring open hole and cased hole measurements with a downhole tool ( 132 ); determining ( 698 ) open hole parameters (e.g., Z90) from the cased hole parameters, known parameters and the open hole measurements; and generating ( 699 ) downhole outputs from the determined open hole parameters.

BACKGROUND

The present disclosure relates to techniques for performing formationevaluation. More particularly, the present disclosure relates totechniques, such as calibrations, that may be used in performingmeasurement, imaging and/or other formation evaluations.

To locate and capture valuable hydrocarbons from subterraneanformations, various wellsite tools may be used to perform various tasks,such as drilling a wellbore, performing downhole testing and producingdownhole fluids. Downhole drilling tools may be advanced into the earthby a drill string with a bit at an end thereof to form the wellbore.Drilling muds (or other drilling fluids) may be pumped into the wellboreand through the drilling tool as it advances into the earth. Thedrilling muds may be used, for example, to remove cuttings, to cool thedrill bit and/or to provide a coating along the wellbore. The drillingmuds may be conductive or non-conductive drilling fluids (e.g., oilbased muds (OBM), water based muds (WBM), etc.) During or afterdrilling, casing may be cemented into place to line a portion of thewellbore, and production tools may be used to draw the downhole fluidsto the surface.

-   -   During wellsite activities, downhole measurements may be taken        to collect information about downhole conditions. The downhole        measurements may be taken of various wellsite parameters, such        as temperature, pressure, permittivity, impedance, resistivity,        gain factor, button standoff, etc. Downhole tools, such as the        drilling tool, a testing tool, a production tool, or other        tools, may be deployed into the wellbore to take the downhole        measurements, such as formation resistivity. These downhole        measurements may be used to generate downhole parameters, such        as impedance of electrodes used in taking the downhole        measurements, vectors of the impedance, and the length of such        vectors (e.g., Z90). In some cases, downhole logs, images or        other outputs may be generated from the downhole measurements.

BRIEF SUMMARY

This disclosure relates to techniques for calibrating downholemeasurements. The techniques involve measurements in cased and open holewellbores that may be used to cross-check sensor measurements and/or todetermine cased and open hole impedances. This information may be usedto calculate Z90 and/or generate calibrated downhole images (or otheroutputs).

In one aspect, the disclosure relates to a method of generatingcalibrated downhole images of a subterranean formation surrounding awellbore. The method involves deploying a downhole tool into a casedportion of the wellbore (the downhole tool having at least one sensorpad for measuring downhole parameters), obtaining cased holemeasurements (e.g., impedance) in a cased hole portion of the wellborewith the sensor pad, determining cased hole parameters from the casedhole measurements, deploying the downhole tool into an open hole portionof the wellbore, obtaining open hole measurements (e.g., impedance) inthe open hole portion of the wellbore with the sensor pad, determiningopen hole parameters (e.g., Z90) from the cased hole parameters, knownparameters and the open hole measurements, and generating (699) downholeoutputs from the determined open hole parameters.

The may also involve determining formation resistivity from Z90, passinga current from at least one button electrode to at least one returnelectrode on the sensor pad and measuring the current, comparing thecurrent measured by a plurality of the button electrode and at least onereturn electrode of the sensor pads, determining a standoff between eachof the button electrode and a measurement surface and adjusting for thestandoff, analyzing the cased hole measurements, the cased holeparameters, the open hole measurements, the open hole parameters and/orZ90 and adjusting therebetween.

The cased hole impedance may include a button impedance and/or a mudimpedance. A measured button impedance may equal a real buttonimpedance. The cased hole impedance has an amplitude, a magnitude, aphase, and an angle. The known parameters may include a casingcurvature, a sensor pad curvature, standoff, and/or mud angle. Acurvature mismatch may exist between the casing curvature and the sensorpad curvature, and adjustments may be made therefor. The cased holeparameters may include a mud angle, a mud permittivity, a standoff, again factor, and/or an amplitude offset. The open hole parameters mayinclude an open hole amplitude and/or an open hole phase.

In another aspect, the disclosure relates to a system for generatingcalibrated downhole images of a subterranean formation surrounding awellbore. The system includes a downhole tool positionable in a casedportion and an open portion of the wellbore. The downhole tool includesat least one sensor pad for measuring downhole parameters, at least onebutton electrode and at least one return electrode on the sensor pad,and electronics in communication with the button electrode and thereturn electrode. The electronics obtain cased hole measurements in acased hole portion of the wellbore and open hole measurements in an openhole portion of the wellbore with the sensor pad, and determine casedhole parameters (e.g., Z90) from the cased hole measurements and openhole parameters from the cased hole parameters, known parameters and theopen hole measurements.

The system may also have at least one guard electrode between the buttonelectrode and the return electrode, at least one wear plate extendingfrom a front fact of the sensor pad, and/or an electrically insulatedmaterial along a front face of the sensor pad with the button electrodeand the electrode positionable therein. A front face of the sensor padhas a curvature with each of the button electrodes positionable alongthe curvature. The sensor pad is positionable against a measurementsurface via at least on leg.

This summary is provided to introduce a selection of concepts that arefurther described below in the detailed description. This summary is notintended to identify key or essential features of the claimed subjectmatter, nor is it intended to be used as an aid in limiting the scope ofthe claimed subject matter.

BRIEF DESCRIPTION OF THE DRAWINGS

Embodiments of methods for calibration a downhole imaging tool aredescribed with reference to the following figures. The same numbers areused throughout the figures to reference like features and components.

FIG. 1-1 illustrates a schematic view, partially in cross-section, of awellsite having a downhole tool with a sensor pads for taking downholemeasurements in which embodiments of methods for calibration can beimplemented.

FIG. 1-2 illustrates a schematic view, partially in cross-section, of aportion 1-2 of the wellsite of FIG. 1-1 depicting one of the sensor padsin greater detail.

FIG. 2-1 illustrates a top view, partially in cross-section, of aportion of the wellsite of FIG. 1-1 taken along line 2 ₁-2 ₁ depictingknown parameters of the wellsite.

FIG. 2-2 illustrates a top view, partially in cross section, of aportion of the wellsite of FIG. 1-1 taken along line 2 ₂-2 ₂ depictingvarious known parameters of the sensor pad.

FIG. 3 is a graph depicting curve fitting of impedance of a plurality ofbutton electrodes.

FIG. 4 is a graph depicting standoff of a plurality of buttonelectrodes.

FIG. 5 is a graph depicting curve fitting of standoff for a plurality ofbutton electrodes.

FIG. 6 is a flow chart depicting an example method of generatingcalibrated images of a wellbore.

DETAILED DESCRIPTION

The description that follows includes example apparatuses, methods,techniques and instruction sequences that embody techniques of thepresent subject matter. However, it may be understood that the describedembodiments may be practiced without these specific details.

The techniques described herein may be used to generate calibrateddownhole outputs, such as downhole images, logs, etc. The methodsinvolve determining downhole parameters, such as impedance and Z90, fromdownhole measurements. The downhole measurements may be taken bydownhole tools with sensor pads positioned in cased and open holeportions of a wellbore. The cased hole and open hole measurements may beused with known parameters (e.g., casing dimensions, sensor paddimensions, etc.) to reduce error that may be caused by measurementvariations, such as raw phase offset, phase button variation, mud angle(sometimes referred to as ‘loss tangent’), amplitude button variation,mud permittivity, sensor variations, curvature mismatch, etc.

FIG. 1-1 is a schematic view of a wellsite 100 having a rig 102positioned over a wellbore 105 penetrating a subterranean formation 110.While the rig 102 in FIG. 1-1 is shown as being land-based, it will beappreciated that the rig 102 could be at an offshore location. Thewellbore 105 may be created using a drilling tool (not shown). Duringdrilling, a drilling mud 112 may be pumped downhole to facilitate thedrilling process. As a result, a layer of mud 111 (or mud cake) may formon a wall 115 of the wellbore 105. The mud 111 may be an oil or waterbased mud. A metal casing 106 may then be cemented along the wall 115with a cement 108, thereby defining a cased hole portion 124 of thewellbore 105 adjacent the casing 106 and an open hole portion 126therebelow.

A downhole tool 114 may be lowered into the wellbore 105 to takedownhole measurements. The downhole tool 114 is depicted as a wirelineimaging tool with sensor pads 117, but may be any downhole tool, such asa micro-imager capable of taking downhole measurements (e.g.,resistivity) in oil or water based mud. The downhole tool 114 may be aconventional resistivity tool used to generate images as described, forexample, in US Patent Application No. 2011/0114309.

As depicted in FIG. 1-1, the downhole tool 114 has a body 132 (ormandrel) with a plurality of spaced arms 116 extending therefrom. Eacharm 116 has the sensor pad 117 operatively attached to an end thereof.One or more sensor pads 117 may be positioned about the downhole tool114, e.g., on arms 116 and/or mandrel 132. The sensor pads 117 may beselectively extendable from the mandrel 132 via the arms 116 for takingdownhole measurements of the formation 110 surrounding the wellbore 105.

The downhole tool 114 may be positioned at various locations along thewall 115 of the wellbore 105 for taking downhole measurements. Thedownhole tool may be lowered into a cased position in the cased holeportion 124 of the wellbore 105 (shown in solid line), and/or into anopen hole position in the open hole portion 126 of the wellbore 105(shown in dashed line). In some cases, the downhole tool 114 may belowered to a calibration position near a bottom of the cased holeportion 124 adjacent the open hole portion 126 of the wellbore. Due tothe proximity between the cased hole portion 124 and open hole portion126, the calibration position may be a location where the downhole tool114 has roughly the same environmental conditions as a top portion ofthe open hole portion 126.

FIG. 1-2 shows a cross-sectional view of the sensor pad 117 of FIG. 1.While the sensor pad 117 is shown as being in a cased hole portion 124and positioned against casing 106, it may be anywhere in the wellbore105 and positionable against any measurements surface (e.g., mud 111,casing 106, wall 115, etc.) The sensor pad 117 is shown having a padbody 118 with an electrode package 141 therein. The pad 117 may have aface 134 positionable along the casing 106 of the wellbore 105. Aportion of the face 134 may be made of an electrically insulatedmaterial 135.

To protect the pad 117 and to keep the face 134 from touching the casing106 (or the wall 115), the sensor pad 117 may be provided with an upperwearplate 130 and a lower wearplate 131 at upper and lower ends,respectively, thereof. The wearplates 130, 131 may protrude a distance(or standoff) d_(b) from the front face 134 to prevent direct contactbetween the front face 134 and the casing 106. When a measurement isdesired, the arms 116 may be selectively extended to a position wherethe pad 117 is flushably pressed against the casing 106 (or the wall115).

The electrode package 141 includes a button electrode 140 at least onereturn electrode 136 and at least one guard electrodes 138. Return andguard electrodes may be on either side of at least one button electrode.The electrode package 141 may optionally include one or more buttonelectrodes 140, return electrodes 136, guard electrodes 138, and/orother electrode capable of taking the desired downhole measurementsthrough drilling mud (oil or water based) and/or formations. Theelectrode package 141 is operatively connected to electronics 142positioned within the pad body 118. While the electronics 142 isdepicted as being in the sensor pad 117, at least a portion of theelectronics 142 may be positioned in the arms 116 and/or the mandrel 132of the downhole tool 114.

As shown in FIG. 1-2, the button electrode 140 emits a current i thatpasses through the drilling mud 111 and through the casing 106 and isreceived by the return electrodes 136. The drilling mud may haveelectrical parameters that may vary over a wide range. Oil-based mudsmay have a relative dielectric permittivity between about 2 and about 40depending, for example, on the oil-water ratio of the mud. A mudimpedance angle of oil-based mud may be in a range of between about −90and about −45 degrees depending on oil-water ratio, temperature andadditives. This mud impedance angle range may correspond to a losstangent between about 0.00 and about 1.00. When in the open holeposition (e.g., 124 of FIG. 1-1), the button electrodes 140 emits acurrent that passes through the drilling mud 111 and/or the formation110.

FIG. 2-1 shows a top view of the downhole tool 114 with four arms 116extended such that four sensor pads 117 are positioned against thecasing 106. One or more arms 116 and sensor pads 117 may be positionedat various positions relative to the downhole tool 114 (evenly orunevenly). Also shown are various downhole geometries, such as thecasing diameter Φ (which may be between about 3 inches (7.62 cm) to 28inches (71.12 cm)), a pad curvature radius φ_(p) and a casing curvatureradius φ_(c) (which may be between about 1.5 inches (3.81 cm) and 14inches (35.56 cm)). These and other known or measureable downholegeometries may be used in the determination of various downholeparameters.

FIG. 2-2 shows a cross-sectional view of one of the sensor pads 117 inthe measurement position against the casing 106. As shown in this view,the sensor pad 117 may have multiple electrode packages 241 along thefront face 134. This view also shows additional downhole geometries,such as a button standoff d_(b1-13), the distance from each set ofelectrodes 241 to the casing 106 (or the wall 115 when in the open holeportion 126 of the wellbore 105 as shown in FIG. 1). The number ofbuttons may vary, for example, as a function of pad width and imageresolution requirements. The button standoff d_(b1-13) for each of theelectrode packages 241 may vary depending on the pad curvature radiusφ_(p) relative to the casing curvature radius φ_(c). A range for thestandoff may be between about 0.10 and 30.00 mm. For standoff valuesoutside this range the quality of the measurement may be degraded. Thestandoff may be within this range for at least a large portion of a logtaken by the downhole tool. For standoff values smaller than about 0.10mm, the measurement may become unstable; for standoff values above about30 mm the resolution and/or sensitivity may degrade.

As shown, the button standoff d_(b1) for the electrode package(s) 241closest to the casing 106 (or wall of the wellbore) may be approximatelythe same as the wear plate standoff d_(b) of the wear plate 131. Due tothe pad curvature radius φ_(p), the button standoff d_(b2-13) for theelectrode packages on either side of the standoff d_(b) increasestowards either end of the pad 117.

The downhole measurements taken by the sensor pads 117 in the cased andopen hole positions and the known downhole geometries may be used todetermine various downhole parameters and/or to generate downhole images(or other outputs). Various factors, such as electrical properties ofthe mud, offsets of the measurement electronics or other measurementfactors, may affect the quality of the measurements and the resultingimages generated therefrom. The methods herein are configured to‘calibrate’ the measurements based on control measurements taken incased hole portion 124 of the wellbore and the factors which may affectmeasurements.

Determining Impedances and Z90

Impedances of the button electrode 140 and the drilling mud 111 trappedbetween the wall 105 of the wellbore and the sensor pad 117 may bedetermined from downhole measurements. The button impedances may be usedto determine various parameters, such as Z90. A vector of the buttonimpedance in a complex plane can be decomposed into two orthogonalvectors: a first vector in the direction of the mud impedance vector,and a second vector orthogonal to the direction of the mud impedancevector. Z90 is a length of the second vector. In some cases, Z90 may beused as a measure of formation resistivity that may be independent ofother parameters, such as standoff and rugosity.

Referring to FIG. 2-2, to obtain the impedances, the sensor pad 117 ofthe downhole tool may be used to pass currents from the button electrode140 through the casing 106 (when in the cased hole portion 124) or theformation 110 (when in the open hole portion 126). A source (e.g.,electronics 142) may generate a voltage between the return electrode 136and the button electrode 140 with at least one spectral component in afrequency range between about a 100 kHz and about 100 MHz. The range ofgenerated voltages may be range from about 0.10 mV to about 1000Vdepending on the required image quality and the resistivity of theformation and the geometry of the pad (e.g., the electrodes). Voltagesbelow the indicated lower limit may result is noisy images, voltagesabove the upper limit may give rise to mud stability problems andexcessive tool power consumption. The voltage between the returnelectrode 136 and the button electrode 140 may result in a measurementcurrent exchange therebetween and, due to the small standoff compared tothe distance between the two electrodes and due to the conductivity ofthe casing 106, the current starts from the button electrode 140,penetrates through the mud 111 layer, and passes into the casing 106.

The current will then run towards the return electrode 136 where thecurrent will leave the casing 106, penetrate through the mud 111 infront of the sensor pad 117 and terminate at the return electrode 136 asindicated by the dashed arrow. As the current is alternating the currentmay run the opposite way or may run both ways at different points intime. The button current may be in the range of between about 10 nA and100 mA. Currents below the indicated lower limit may result is noisyimages, and currents above the upper limit may give rise to excessivetool power consumption.

The button current together with the return-button voltage, defines abutton impedance (Z_button):Z_button=V_return/I_button[Ohm]  Equation (1)The equations herein may be identified as being based on variousparameters, such as resistance [Ohms], voltage [V], current (or amps)[A], capacitance [F], and frequency [rad]. The button impedance(Z_button), the return voltage (V_return) and the button current(I_button) are complex phasors having a magnitude and a phase shiftwhich can be expressed as:Z_button=abs(Z_button)*exp(i*angle(Z_button))[Ohm]  Equation (2)V_return=abs(V_return)*exp(i*angle(V_return))[V]  Equation (3)I_button=abs(I_button)*exp(i*angle(I_button))[A]  Equation (4)where the absolute value for Z_button, V_button and I_button arefunctions that give the magnitude of the signal for Z_button, V_buttonand I_button, and the angle for Z_button, and V_button and I_button is afunction that gives the phase angle of the signal for Z_button, V_buttonand I_button (e.g. in the interval [−pi, pi]). Angle ( ) may be based ona function y=angle(x) which is defined as the function y=a tan2(imag(x),real(x)) where the a tan 2( ) function is defined in Organick,Elliott I, A Fortran IV Primer, Addison-Wesley. pp. 42 (1966). Someprocessors may also offer a library function called ATAN2, a function oftwo arguments (opposite and adjacent). The function imag( ) gives animaginary part of the complex variable x and the function real( ) givesa real part of the complex variable x. The function y=abs(x) may bedefined a function y=sqrt(imag(x)2+real(x)2), where sqrt( ) denotes theusual square root function. Written out as full equations the followingmay be provided:Z_button=V_return/I_button[Ohm]  Equation (5)=abs(V_return)*exp(i*angle(V_return))/abs(I_button)*exp(i*angle(I_button))[Ohm]  Equation(6)=abs(V_return)/abs(I_button)*exp(i*{angle(V_return)−angle(I_button)})[Ohm]  Equation(7)which shows that the button impedance magnitude is the ratio of thereturn voltage amplitude and the button current amplitude, while thebutton impedance phase is the difference between the return voltagephase and the button current phase as shown in Equations (8) and (9):abs(Z_button)=abs(V_return)/abs(I_button)[Ohm]  Equation (8)angle(Z_button)=angle(V_return)−angle(I_button)[rad]  Equation (9)Z_button may be considered to be a measurement taken by, for example,the sensor pad 117 and/or downhole tool 114.

Z90 may be determined from the impedances while taking into accountpotential sensitivities which may cause measurement error. For example,if the mud impedance angle is incorrectly estimated, then the Z90quantity can become sensitive to the standoff d_(b). If Δangle(Z_mud) isthe estimated mud impedance angle minus the real mud impedance, then:Δangle(Z_mud)=angle_estimated(Z_mud)−angle(Z_mud)[rad]  Equation (10)Performing a similar calculation as above generates the following:Z90=abs(Z_button)*sin[angle(Z_button)−angle_estimated(Z_mud)][Ohm]  Equation (11)which shows thatZ90=abs(Z_mud)*sin {−Δangle(Z_mud)}+abs(Z_form)*sin{angle(Z_form)−angle(Z_mud)−Δangle(Z_mud)}[Ohm]  Equation (12)The above equations show that, given some difference between theestimated mud angle and the real mud angle, the Z90 quantity can becomesensitive to the amplitude of the mud impedance abs(Z_mud) since thefirst term on the right is not zero. Because the amplitude of the mudimpedance may be dependent on standoff, the Z90 quantity can also bedependent on standoff. Thus, an estimation of the mud angle(angle(Z_mud)) may be used in the determination of Z90.

Z90 may also be affected by incorrect measurement phase angle. If eitheror both of the phase of the return voltage measurement and the phase ofthe button current measurement have an error or offset, then the phaseangle of the button impedance (angle(Z_button)) can have an error(assuming the two errors do not cancel). The effect on the measure Z90may be the same whether an offset or error is in one of the other orboth (except for the sign). Assuming the following:Δangle(Z_button)=angle_measured(Z_button)−angle(Z_button)[rad]  Equation(13)then it follows that:Z_button_measured=abs(Z_button)*exp{i*angle_measured(Z_button)}[Ohm]  Equation(14)=abs(Z_button)*exp{i*angle(Z_button)}*exp{i*Δangle(Z_button)}[Ohm]  Equation(15)=Z_button*exp{i*Δangle(Z_button)}[Ohm]  Equation (16)=(Z_mud+Z_formation)*exp{i*Δangle(Z_button)}[Ohm]  Equation (17)Performing a similar calculation Z90 may be generated as follows:Z90=abs(Z_button)*sin[angle_measured(Z_button)−angle(Z_mud)][Ohm]  Equation (18)Z90 may be rewritten as follows:Z90=abs(Z_mud)*sin {Δangle(Z_button)}+abs(Z_form)*sin{angle(Z_form)+Δangle(Z_button)−angle(Z_mud)}.[Ohm]  Equation (19)This shows that, given some difference between the measured phase of thebutton impedance and the real phase of the button impedance, the Z90quantity can become sensitive to the amplitude of the mud impedanceabs(Z_mud) since the first term on the right is not zero. Because theamplitude of the mud impedance may be dependent on standoff, the Z90quantity may also be dependent on standoff. Thus, the measured phase ofthe button impedance may need to be the same or close to the real phaseof the button impedance.

Z90 may be dependent on incorrect mud angle and incorrect measurementphase angle. By combining the impedance and phase angle equations aboveand the effect of both, an error or offset in the mud angle and an erroror offset in the measured button impedance phase may be derived.Assuming the following:Z90=abs(Z_button)*sin[angle_measured(Z_button)−angle_estimated(Z_mud)][Ohm]  Equation (20)it follows that:Z90=abs(Z_mud)*sin {Δangle(Z_button)−Δangle(Z_mud)}+abs(Z_form)*sin{angle(Z_form)−angle(Z_mud)+Δangle(Z_button)−Δangle(Z_mud)}[Ohm]  Equation(21)If Δangle(Z_button)−Δangle(Z_mud)=0 (i.e., if the error or offset in themeasured phase is equal to the error or offset in the estimated mudangle such that the difference is zero), then Z90 may be independent ofthe amplitude of the mud impedance and, therefore, in first orderapproximation, independent of standoff.

Z90 may also be used to determine various formation parameters. Theformation properties may be derived from the Z_button measurements. Thebutton impedance (Z_button) may be used as a first order approximationequal to the sum of the impedance of the path through the mud betweenthe button electrodes and the formation, and the impedance of a paththrough the formation itself. In other words, the button impedance maybe the total impedance of the mud impedance and the formation impedancein series where the impedances are complex phasors as described aboveand as set forth below:Z_button=Z_mud+Z_formation[Ohm]  Equation (22)The mud impedance between the button electrode and the formation maydominate the mud impedance between the return electrode and theformation. Depending on model accuracy, the latter impedance may beneglected. The formation properties may then be obtained from Z_button.Z_mud may be estimated and subtracted from Z_button as follows:Z_formation=Z_button−Z_mud.[Ohm]  Equation (23)Z_fomation gives the complex formation impedance which can be convertedwith a k-factor (geometric factor) into a resistivity and a permittivityof the formation.

The complex mud impedance (Z_mud) may be difficult to estimate since itvaries with the thickness of the mud layer between the button electrodeand the formation (standoff). To address this, the phase angle of themud impedance may be used as follows:Z90=abs(Z_button)*sin [angle(Z_button)−angle(Z_mud)][Ohm]  Equation (24)=imag(abs(Z_button)*exp{i*[angle(Z_button)−angle(Z_mud)]})[Ohm]  Equation(25)=imag(abs(Z_button)*exp{i*angle(Z_button)}*exp{−i*angle(Z_mud)})[Ohm]  Equation(26)=imag([abs(Z_mud)*exp{i*angle(Z_mud)}+abs(Z_form)*exp{i*angle(Z_form)}]*exp{−i*angle(Z_mud)})[Ohm]  Equation(27)=imag([abs(Z_mud)+abs(Z_form)*exp{i*angle(Z_form)−angle(Z_mud)})[Ohm]  Equation(28)=abs(Z_form)*sin {angle(Z_form)−angle(Z_mud)}[Ohm]  Equation (29)This shows that, with the help of an estimated mud impedance phase angleangle(Z_mud), a quantity Z90 can be determined from the measurement. Z90may be approximately proportional to the magnitude of the formationimpedance Z_formation when there is a difference between the phase angleof the mud impedance and the phase angle of the formation impedance.

Z90 may be independent of the amplitude of the mud impedance abs(Z_mud)and, therefore, independent of the thickness of the mud layer betweenthe button electrode and the formation (standoff). The formationresistivity may be over a large range of resistivity valuesmonotonically related to the magnitude of the formation impedance and,therefore, to the Z90 quantity. Z90 can, thus, be used as a measure offormation resistivity. Based on the above, from the measured buttonimpedance (Z_button) and an estimation of the mud angle (angle(Z_mud)),a quantity Z90 which may be a metric/measure of formation resistivitycan be determined. This metric/measure may be independent of standoff.

In some cases, it may be useful to adjust Z90 for various errors thatmay occur under various conditions, such as mud angle, formationresistivity, standoff, amplitude of mud impedance, phase angle, andphase button variation. By determining various known and measuredparameters, Z90 may be adjusted. For example, an incorrect mud angle (ormud impedance angle) can affect Z90. If the mud angle is incorrectlyestimated, then the Z90 quantity may become sensitive to the standoff.Where Δangle(Z_mud) is the estimated mud impedance angle minus the realmud impedance angle, it follows that:Δangle(Z_mud)=angle_estimated(Z_mud)−angle(Z_mud)[rad]  Equation (30)Performing a similar calculation as above, Z90 can be expressed asfollows:Z90=abs(Z_button)*sin[angle(Z_button)−angle_estimated(Z_mud)][Ohm]  Equation (31)Z90=abs(Z_mud)*sin {−Δangle(Z_mud)}+abs(Z_form)*sin{angle(Z_form)−angle(Z_mud)−Δangle(Z_mud)}[Ohm]  Equation (32)Based on the above equations, given some difference between theestimated mud angle and the real mud angle, the Z90 quantity may becomesensitive to the amplitude of the mud impedance abs(Z_mud) since thefirst term on the right is not zero. Because the amplitude of the mudimpedance may be dependent on standoff, the Z90 quantity may also bedependent on standoff. Thus, an estimation of the mud angle(angle(Z_mud)) may be useful.

Measurement phase angle may affect Z90. If either or both of the phaseof the return voltage measurement and the phase of the button currentmeasurement have an error or offset, then the phase angle of the buttonimpedance (angle(Z_button)) may have an error (assuming the two errorsdo not cancel). The effect on the measure Z90 may be the same whetherthe offset or error is in one of the other or both (except for thesign). Assuming the following:Δangle(Z_button)=angle_measured(Z_button)−angle(Z_button)[rad]  Equation(33)it follows that:Z_button_measured=abs(Z_button)*exp{i*angle_measured(Z_button)}[Ohm]  Equation(34)=abs(Z_button)*exp{i*angle(Z_button)}*exp{i*Δangle(Z_button)}[Ohm]  Equation(35)=Z_button*exp{i*Δangle(Z_button)}[Ohm]  Equation (36)=(Z_mud+Z_formation)*exp{i*Δangle(Z_button)}[Ohm]  Equation (37)and performing a similar calculation as above with the following:Z90=abs(Z_button)*sin[angle_measured(Z_button)−angle(Z_mud)][Ohm]  Equation (38)it follows that:Z90=abs(Z_mud)*sin {Δangle(Z_button)}+abs(Z_form)*sin{angle(Z_form)+Δangle(Z_button)−angle(Z_mud)}[Ohm]  Equation (39)This shows that, given some difference between the measured phase of thebutton impedance and the real phase of the button impedance, the Z90quantity may become sensitive to the amplitude of the mud impedanceabs(Z_mud) since the first term on the right is not zero. Because theamplitude of the mud impedance may be dependent on standoff, the Z90quantity may also be dependent on standoff. Thus, it may be useful forthe measured phase of the button impedance to be the same or close tothe real phase of the button impedance.

The mud angle and phase angle may also affect Z90. The mud angle andmeasured phase angle can be combined to derive the effects of error oroffset in the mud angle and error or offset in the measured buttonimpedance phase. Assuming the following:Z90=abs(Z_button)*sin[angle_measured(Z_button)−angle_estimated(Z_mud)][Ohm]  Equation (40)it follows that:Z90=abs(Z_mud)*sin {Δangle(Z_button)−Δangle(Z_mud)}+abs(Z_form)*sin{angle(Z_form)−angle(Z_mud)+Δangle(Z_button)−Δangle(Z_mud)}[Ohm]  Equation(41)If Δangle(Z_button)−Δangle(Z_mud)=0, or if the error or offset in themeasured phase is equal to the error or offset in the estimated mudangle such that the difference is zero, then Z90 may be independent ofthe amplitude of the mud impedance and, therefore, provide a first orderapproximation independent of standoff.Cased Hole Analysis

The cased hole measurements, such as those taken using the downhole tool114 in the cased portion 124 of the wellbore 105 of FIG. 1-1, may beused to provide reference measurements in calibrations or as inputs forfurther analysis. In using the cased hole measurements, one or more ofthe following can be assumed: 1) the casing 106 is a near perfectconductor or insulator, 2) the cased hole measurements have a comparablemagnitude of measurement current in a low-resistivity formation, 3) thecased portion 124 has similar environmental conditions as the adjacentopen hole portion 126, 4) the casing 106 has a known diameter Φ andcurvature radius φ_(c) (FIGS. 2-1), and 5) the casing 106 has a smoothsurface with negligible rugosity.

The phase angle of the measurement (e.g. return voltage divided bybutton electrode current) may be defined by the phase angle of the mudand the phase offset of the measurement system. A basic two impedancemodel of the current-injection measurement may be used. A voltage isapplied across two complex impedances in series, one representing themud the other representing the formation (or the casing if the tool islocated in the casing). The current is measured and the voltage dividedby this measured current to generate the total impedance of the twoimpedances in series. If the formation/casing impedance is nearly zero(due to the high casing conductivity), then the current may be definedby the mud impedance. Thus, the total impedance may be assumed to beequal to the mud impedance. The magnitude of the mud impedance may varywith the standoff (thickness of the mud layer between the casing and theelectrodes). Thus, the mud impedance phase angle may not vary with thestandoff as it is assumed to be geometry invariant.

Various techniques may be performed, such as comparisons, to evaluatemeasurements and address various issues affecting the measurement. Forexample, the cased measurements may be used to ensure that the referencecasing measurement has roughly the same measurement current magnitude asthe open-hole formation measurement. Electronic measurement systems maybehave differently for different signal magnitudes. When measuring aweaker signal, the effect of internal crosstalk may be different thanwhen measuring a stronger signal. In another example, a weakermeasurement may be performed at different electronic gain settings withdifferent gain and phase offsets. The signal strengths may be similarbetween a measurement taken in the casing as a reference measurement anda measurement of a low resistivity formation. Therefore, application ofa casing reference measurement to correct the formation measurement maybe less prone to errors than corrections based on reference measurementsthat have a different signal strength than the strength of the signal tobe corrected.

Mud properties can also be measured by measuring with the sensor padsretracted from the formation (i.e. with a reference pad closedmeasurement). However, this measurement may involve weaker signals thanwhen the sensor pads are pressed against a borehole wall with a relativelow formation resistivity. In another example, the cased measurementsmay also address variations in temperature and pressure of boreholefluids. Drilling mud can be measured at surface and tools can becalibrated at surface as well. To obtain the desired mud correction,however, these surface measurements may be extrapolated to downholepressure and temperature conditions, or measured at surface undersimilar pressure and temperature conditions as downhole.

In yet another example, the cased measurements may also be used toaddress curvature of the sensor pad relative to the casing (or padcurvature radius φ_(p)) as shown in, for example, in FIG. 2-2. If anarray of electrodes is placed on a sensor pad with a known curvature andif this pad is pressed against the casing, then the signal amplitudesthe other electrodes can be determined based on measurements of a fewelectrodes, (e.g., 2 or 3 electrodes).

The obtained casing measurements may be used to generate variousreference measurements (or cased hole parameters), such as raw phase,phase button variation, phase calibration, mud angle, amplitude buttonvariation, amplitude calibration, and mud permittivity. The casingmeasurements may be, for example, resistivity measurements taken incasing to generate one or more image measurements and one or more returnvoltages using one or more return electrodes in a sensor pad asdescribed above.

Raw phase may involve a determination of a raw phase offset (includingmud angle effect and electronics phase offset) to be used withorthogonal processing. The phase of measured impedance is the raw phaseoffset in the Z90 processing including both the mud angle and the sensorphase correction.

Phase button variation may involve a reduction in phase variationbetween multiple button electrodes (see, e.g., FIG. 2-2). In this case,voltage and current may be used to determine a phase of measured buttonimpedances. Because the casing is assumed to be a near perfectconductor, the buttons may read the same phase (in principle equal tothe phase of the mud impedance). Variations in readings in the phase ofthe measured button impedances may be calibrated (or corrected) withrespect to each other.

Phase calibration may involve calibration of a sensor using a known mudangle. One or more return and one or more button electrodes may be usedto generate a voltage and current. This information may be used togenerate a phase offset of the measured impedance. This offset may beused to correct the sensor or measurements taken by the sensor toprovide a more accurate measurement.

The mud angle may be known, or determined from other measurements.Because the casing may be assumed to be a near perfect conductor, themud impedance may determine the phase of the measured impedance. Thephase of measured impedance may be equal to the mud angle. Thus, thephase of the measured impedance may be calibrated (or corrected) usingthe mud angle.

A determination of mud angle may be performed. One or more button andreturn electrodes may be used to generate voltage and current. Thisinformation may be used to generate a phase of measured impedance.Because casing may be assumed to be a near perfect conductor, the mudimpedance may determine the phase of the measured impedance and,therefore, the mud angle may be equal to the phase of measuredimpedance. Thus, the mud angle can be converted into an equivalent losstangent tan(delta_mud).

Amplitude button variation involves reducing amplitude variationsbetween buttons using a fitting curve or a button standoff (e.g.,d_(b1-13) of FIG. 2-1). One or more return electrodes may be used withmultiple button electrodes to generate multiple button currents. Thevoltage and button currents may be used to generate measured buttonimpedances. These may be fitted to a smooth curve (e.g., quadratic)through the impedances. Mismatch between the curve and button impedancesmay be determined for each button electrode. The mismatches may be usedto correct button impedances and to reduce button to button impedancevariations.

Amplitude calibration involves calibration of sensors using mudpermittivity and wear plate standoff d_(b) (FIG. 2-1). One or morereturn electrode, one return voltage, one image button, and one buttoncurrent may be used to generate voltage and current. A measuredimpedance may be generated from the voltage and current.

Mud permittivity may be known, or generated from other measurements. Themud permittivity may be processed with estimated standoff, button size,signal spectrum (frequency), and impedances calculated. The calculatedimpedance may be used to correct or calibrate the measured impedancecalculated from the voltage and current. Mud permittivity may beestimated using button impedance and wear plate protrusion. For example,one or more return electrodes, one return voltage, one image button, andone current button may be used to generate voltage and current. Measuredimpedance may be generated from the voltage and current. The impedancemay be processed with estimated standoff, button size, parallel platecapacitor model, signal spectrum (frequency), and epsilon mud (∈mud) maybe generated.

The cased hole measurements may also be used in combination withcurvature mismatch concepts, such as raw phase curvature, mudpermittivity and button standoff, and amplitude calibration curvature(discussed further below). Raw phase curvature involves a determinationof a raw phase offset (including mud angle effect and electronics phaseoffset) to be used with orthogonal processing. One or more returnelectrodes may be used to generate one return voltage. A differencebetween a casing curvature radius φ_(c) and a pad curvature radius φ_(p)may be known. Two image buttons may be chosen such that these havedifferent standoffs due to curvature difference. A voltage and currentmay be generated from the button electrodes, and a phase of measuredimpedance generated. Z90 may be determined from processed data, and acasing resistivity generated. Raw phase offset of measured data may beoptimized until Z90 processing gives a casing resistivity equal to about0.00 Ohms and/or negligible variation between button electrodes withdifferent standoff. The sum of the mud angle and the sensor phasecorrection may be used as an optimal phase offset.

Mud permittivity and button standoff may be performed using curvaturemismatch between pad and casing (difference between a casing curvatureradius φ_(c) and a pad curvature radius φ_(p)). One or more returnelectrodes may be used to generate a return voltage. A casing radius,pad curvature radius and pad geometry with button layout may be known ormeasured. For example, three or more image buttons may be chosen suchthat these have different standoff due to curvature difference, andthree button current measurements may be generated. The standoffdifference may be calculated based on geometry (positions of buttons,curvatures, estimated positioning of the pad against casing (whichbutton is closest to the casing)). Voltage and currents may bedetermined, and measured impedance generated. The information may beprocessed (measured impedance difference, standoff difference, buttonsizes, signal spectrum (frequency)), and epsilon mud and standoff ofindividual buttons may be determined.

Amplitude calibration of sensors using differential standoff instead ofestimated standoff may be performed. This may involve a combination ofmud permittivity standoff curvature and amplitude calibration asdescribed above.

The cased measurements may be used to estimate the mud angle and themeasurement phase offset, for example, in a single step. As shown inFIG. 2-1 through 2-2, the sensor pad 117 may be positioned against thecasing with standoff elements (e.g. wear plates 130, 131 or hardinsulation protruding elements on the pad) preventing the button andreturn electrodes 140, 136 from touching the metal casing 106.Therefore, a layer of mud 111 may remain between the button electrode140 and the metal casing 106, the latter of which may have a resistivityat least two orders of magnitude lower than the formation. In oneexample, raw phase offset may be used to estimate mud angle in thecasing. The raw phase offset (RawPhs) (including mud angle effect andelectronics phase offset) may be determined for use with orthogonalprocessing.

A measurement of the phase of the return voltage and of the phase of thebutton current in the casing may be performed. The phase of the buttonimpedance, angle_measured(Z_button) in the casing, may be determined bytaking the difference of the two phase values. Using equation (22),Z_formation in this case represents the impedance of the current paththrough the casing and Z_mud represents the impedance of the currentthrough the mud. Because the casing is assumed to be a good conductor,Z_formation is assumed to be very close to zero and it can be neglectedsuch that in the casing may be represented as follows:Z_button_casing≈Z_mud_casing[Ohm]  Equation (42)which gives:angle(Z_button_casing)≈angle(Z_mud_casing)[rad]  Equation (43)and, thus:angle_measured(Z_button_casing)≈angle(Z_mud_casing)+Δangle(Z_button)[rad]  Equation(44)Using Equation (10) above and selecting angle_measured(Z_button_casing)as the estimated mud angle, the following is generated:angle_estimated(Z_mud)=angle_measured(Z_button_casing)[rad]  Equation(45)which may be rewritten as:Δangle(Z_button)≈Δangle(Z_mud)[rad]  Equation (46)Such that, in the following formula:Z90=abs(Z_mud)*sin {Δangle(Z_button)−Δangle(Z_mud)}+abs(Z_form)*sin{angle(Z_form)−angle(Z_mud)+Δangle(Z_button)−Δangle(Z_mud)}[Ohm]  Equation(47)the first term becomes negligible and, therefore, Z90 may becomestandoff independent. In other words, if the button impedancemeasurement has a phase offset, then the button impedance measurement inthe casing may also have the same phase offset because it is measuredwith the same system. Then, if the casing measurement is used as ameasurement of the mud angle, then the mud angle may have about the samephase offset as the standard openhole measurement. In the Z90 formula,the mud angle can be subtracted from the open hole button impedanceangle and, therefore, if both contain the same offset (or error), theoffset disappears as set forth below:(A+offset)−(B+offset)=A−B[rad]  Equation (48)

Based on the above methodology, the cased hole analysis may be used toverify sensor measurements and determine various cased hole parameters(e.g., mud, tool or other parameters measured in the cased holeparameters). The downhole tool 114 may have several imaging buttonswhere neighboring button electrode measurements correspond toneighboring pixels in the borehole image. An image processing techniqueknown as equalization can be done after the data has been converted to aresistivity image. This technique may be used, for example, inconfigurations where the image buttons measurements have varied averagevalues over a chosen equalization window length, or where the imagebuttons give the same response in front of the same formation. If theimage buttons respond differently, the final image may have darker andlighter vertical lines that may hide relevant information in the image.

Casing measurements may be used to help align the image buttons torespond in the same way under the same conditions where the electronicsand sensor phase offsets cancel using the phase button variation asdiscussed above. Thus, for each image button the offsets may cancel andthe image buttons may read the same Z90 value in front of the sameformation.

In addition, the casing measurement may give an indication of relativephase offsets of the different image buttons. The measurements of theindividual image buttons in the casing each can be identified with theangle of the mud impedance as described above. The fact that the mud isthe same in front of different buttons may suggest that the phase ofeach button measured should read the same where no phase offsets arepresent in the button measurements (due to sensor or electronicsoffsets), or if the phase offsets of the image buttons were the same. Ifthe image buttons have different phase offsets, then an average andestimate of which buttons have a small or large offset with respect tothe average can be determined. In some cases, individual image buttonsmay have such large offsets that some measurements may be replaced withdata from a neighboring button(s), either by, for example, averaging thedata of the buttons on either side.

By performing casing measurements in subsequent well-logging runs orjobs, the relative button phase offsets may be evaluated to trackwhether the tool degrades and/or varies over time and under differentknown conditions. In other words, phase offsets may act as an indicatorthat can be used for quality control.

In some Z90 applications, the measured phase values may be correctedsuch that they read the same in the casing as is expected. Bydetermining the difference between each button phase and average phase,the button offset may be determined. The button phase values may becorrected by subtracting each button offset from the correspondingbutton phase.

Open Hole Analysis

Open hole analysis may be performed using many of the same techniques asthe cased analysis. The downhole tool 114 may be positioned in the openhole portion 126 of the wellbore as shown in FIG. 1-1 for performingopen hole measurements. The open hole measurements may be performedusing mud measurements with the same electronics and the same sensors asthe cased hole measurements such that the phase offsets due toelectronics and sensors both cancel. Measurements and analysis based oncased hole and open hole applications (as well as known parameters) maybe evaluated.

The casing measurement (AngMudpOff), an open hole amplitude measurement(AbsZpOff), and an open hole phase measurements (AngZpOff) may be usedto determine Z90 using the following equation:Z90=AbsZpOff*sin [AngZpOff−AngMudpOff][Ohm]  Equation (49)The resulting Z90 may be a metric or measure of formation resistivitynearly independent of standoff. An image may be generated from theresulting Z90.Separation of Measurement Offsets and Mud Effects

In some cases the mud angle may be known, or measured with another tool,such as a dielectric scanner or other downhole tool. It may also bepossible to measure at surface and the value downhole derived. In suchcases, the measurement may be done in the casing with the imaging tool.The measured button impedance phase may read the mud angle. If not, thephase measurement can be calibrated (or corrected) by applying adifference between an expected measurement and an actual measurement toeach button phase.

In some cases, the calibration of the tool may be such that the offsetsdue to electronics and sensor offsets may be small or sufficientlyaccurately known in comparison with the accuracy with which the mudangle is determined. For example, if the offsets are smaller than about1 degree and a button impedance angle of about −80 degrees is measured,the mud impedance angle may lie between about −79 and about −81 degrees.Thus, the mud angle can be determined assuming that the known offsetsare sufficiently accurate.

The mud angle may be used as an indication of the quality or stabilityof the mud. Where the mud angle value is above −90, then the more likelythe mud angle may vary over different parts of the open-hole log, themore likely the mud will be inhomogeneous, and the more likely the anglebetween the formation impedance angle and the mud angle will be small.Thus, a degraded oil-based mud imaging performance for mud angles farabove −90 deg can be expected.

Amplitude

In some cases, the casing measurement may not be sufficient as areference for button impedance phase measurement, or as a reference forbutton impedance amplitude measurement. In a first order approximation,the button impedance in the casing may be equal to the mud impedanceassuming the casing is a good conductor such that the ‘formation’impedance is negligibly small. The mud impedance may then beapproximated as the impedance of a (leaky) parallel plate capacitor. Thecomplex capacitance formula for such a capacitor is:C=A*(eps_r*eps0+(sigma/(i*omega)))/s[F]  Equation (50)where C is the capacitance, A is the button area, eps_r is the relativemud permittivity, eps0 is the permittivity of free space, i is thesquare root of −1, omega=2*pi*operating frequency, sigma=the mudconductivity, and s=the mud thickness between the button and the casing(standoff). The complex mud impedance then follows as:Z_mud=(i*omega*C)⁻¹ =s/(i*omega*A*eps_r*eps0+A*sigma)[Ohm]  Equation(51)The button area (A) or any other parameter may be slightly adjustedcompared to reality to include fringing capacitance or other geometriceffects, such as the casing and the button not being parallel or eitherbeing curved.

A measurement in the casing with a known-geometry tool provides ameasured impedance amplitude: abs_measured(Z_button_casing). Thisimpedance measurement can be used to solve for one out of three mainunknowns defining the measured button impedance amplitude: gain factor(p), button standoff (s) and the relative mud permittivity (eps_r). Theone unknown can be solved if information about two other unknowns can bedetermined. The mud conductivity may to first order be neglected.

The measured amplitude may have an offset with respect to the realimpedance amplitude. A gain factor p may be defined as follows:abs_measured(Z_button_casing)=p*abs(Z_button_casing)[Ohm]  Equation (52)The factor p may change over time, with temperature, pressure, etc. Thefactor p can originate from gain offsets in the measured current and/orvoltage. The three unknowns enter in the equation in a linear manner asfollows:abs_measured(Z_button_casing)=cnst*p*s/eps_r[Ohm]  Equation (53)where cnst is some known constant. Depending on other informationprovided, there are several optional approaches: 1) determining p, giveninformation about s and eps_r or information about s/eps_r, 2)determining s, given information about p and eps_r or information aboutp/eps_r, 3) determining eps_r, given information about p and s orinformation about p*s, 4) determining p*s, given information abouteps_r, 5) determining s/eps_r, given information about p, 6) determiningp/eps_r, given information about s, and/or 7) verifying whetherinformation about p, s and eps_r is coherent.

Information may be based on the order of magnitude, an interval ofconfidence, a value with some standard deviation, or some other basis.An example of information about p involves a calibration measurement ofthe tool at the surface. A standard calibration setup involves placingthe sensor pad at a given small distance (e.g., about 4 mm) from a metalsheet (which may be curved to follow the shape of the pad). A (known)gas or fluid or solid material may be placed between the pad and thesheet. The standoff and the medium between the button and the metalsheet may be accurately known or measured. Then a tool measurement maybe taken. From this measurement and knowledge of the standoff andmedium, factor p can be determined. This p can be used later in thedownhole measurement as input information.

As shown in FIG. 2-2, button standoffs d_(b1-13) and the protrusion ofthe wear plates 130, 131 (or standoff d_(b)) of the pad 117 is known.The button-to-casing distance (or standoff) may be assumed to be closeto the wear plate protrusion (or standoff). For example, eps_r(Equations 50-53) is a value obtained with another measurement, such asa dielectric scanner or a separate mud sensor. An example of informationabout p/eps_r is a value obtained by measurement with the sensor pads adistance from the casing or borehole wall. If both p and are known, forexample from the above examples, then the mud permittivity downhole canbe determined. If both eps_r and s are known, for example from the abovetwo examples, then the system amplitude offset (p) downhole can bedetermined.

Multiple Buttons

In the case of a tool with an array of button electrodes, an impedancemeasurement can be generated for each button electrode. The buttonelectrode may be indexed through j for n measurements as follows:abs_measured(Z_button_casing)_j=cnst_j*p_j*s_j/eps_r[Ohm]  Equation (54)where eps_r is not a function of j because the mud permittivity may bethe same in front of the buttons in the casing, cnst_j is known for eachbutton, and the default value for p_j is one.

Although a curvature difference between the button array and the casingmay lead to a different standoff s for each button electrode, thestandoff s may not be random. The button electrodes may follow a part ofa circle with a curvature radius r1. This part of the circle will be insome way defined by a mandrel position, articulation, wear plates, etc.,inside the circular casing with inside curvature radius r2 (which isalso known) (see, e.g., φ_(c) of FIG. 2-1).

The standoffs of the array of button electrodes starting on one edge ofthe sensor pad 117 towards the other edge of the sensor pad may be asmooth curve. For example, if the pad curvature radius φ_(c) is smallerthan the casing radius Φ_(c), then the button electrodes at the edgewill have the most standoff (e.g., d_(b7), d_(b13)). The standoffreduces smoothly to the minimum value as the buttons are followedtowards the middle of the sensor pad (e.g., d_(b1)) and assuming acentered mandrel and standard sensor pad articulation. Thus, a smoothcurve of standoffs may be assumed as a function of button electrodepositions.

The gain factors p_j may not follow a smooth curve. They may have randomoffsets. These random offsets can be corrected by plotting the functionabs_measured(Z_button_casing)_j as a function of j, and then changingthe values p_j until abs_measured(Z_button_casing)_j becomes a smoothfunction of j. For example, abs_measured(Z_button_casing)_j may beplotted as a function of j for the default values of p_j and fit aquadratic curve through the points. The distances between the individualpoints and the curve can then be translated into the following valuesfor p_j:p_j=abs_measured(Z_button_casing)_j/quadratic_j[Ohm]  Equation (55)where quadratic_j means the value of the quadratic fitting function atposition of button j.

Button electrodes having a value for p_j that is different from the restof the button electrodes can also be detected. This may be an indicationof a weak or broken button which may affect images, for exampleresulting in a striping effect in the final image after processing. Thedata may be replaced or enhanced as previously explained.

Curvature Mismatch

If there is a known curvature mismatch between the pad curvature φ_(c)and the curvature of the inside of the casing Φ_(c) (see, e.g., FIG.2-1) then, in the case of multiple buttons, the eps_r of the mud on onehand and a value for standoff for each button electrode on the otherhand can be separated. Thus, mud permittivity can be distinguished fromstandoff. The standoff difference that various buttons may have in orderto respect geometrical constrains of pad and casing curvature can beestimated.

In an example implementation, the sensor pad 117 may be pressed againstcasing 106 as shown in FIG. 2-2. Some additional standoff keepersprevent the button electrodes from touching (e.g., protruding wearplates). For example, six button electrodes with button current spacedat roughly 10 mm may be provided. The button impedance (Zibut) may bedetermined as some voltage divided by the button current for each buttonelectrode. A curve may be fitted through the button impedance values.This curve may have a minimum at the button position where the buttonelectrode is closest to the casing. In other words, the position of theminimum of the fitting curve determines where the sensor pad is closestto the casing. This information together with the geometry of the sensorpad and the curvature of the casing, may provide an estimate of therelative casing standoff of each button.

For a minimum distance (or standoff) (x-axis) between the sensor pad andthe casing indicated by the variable somin (i.e. the standoff atposition 3.4 of FIG. 3 shown below) and assuming known pad curvature,known button electrode positions on the pad and known casing curvature,the button standoff for each button can be determined. For a function(fibut) giving the standoff for each button electrode in a casing withdiameter (casingdiam), for a sensor pad with a curvature radius(crvrad), and for a given value somin at a minimum standoff position(xsomin), the standoff for each button (soibut) is given by:soibut(somin)=fibut(casingdiam,crvrad,somin,xsomin)[m]  Equation (56)

FIG. 3 is a graph 300 depicting button impedances (Y-axes) versus buttonelectrode position (x-axis). The button impedances 370 are depicted witha quadratic polynomial fit line 372 therethrough. The quadratic fitshows a minimum at button electrode position 3.4, i.e. the azimuthposition closest to the formation, measured in button step units, islocated at position xsomin=3.4 units. Based on FIG. 3, the following maybe approximated:soibut(somin)≈fibut(casingdiam,crvrad,0,xsomin)+somin[m]  Equation (57)In this case there is maximum a few percent error at the edge buttonelectrodes given that somin less than or equal to about 5 mm. FIG. 4 isa graph 400 depicting a small difference between soibut (somin) 474 andthe approximation fibut(casingdiam, crvrad, 0, xsomin)+somin 476, forsomin=5 mm. In this graph 400, standoff (y-axis) is plotted againstbutton position (x-axis).

A simple model may relate the button impedance to the button standoff.The model may be based on an approximation of the button impedance incasing by a parallel plate capacitor where the button surface is oneplate and the casing is the other plate. The mud in between may act as a(lossy) dielectric and, therefore, in first order approximation:Zibut=1/(i*ωCibut),[Ohm]  Equation (58)with:Cibut=∈mud*∈0*Abut/soibut,[F]  Equation (59)where ω is the angular frequency, ∈0 is the free-space dielectricpermittivity, ∈mud is the relative mud dielectric permittivity and Abutis the button surface area. From this, it may be deduced that the buttonimpedance is approximately proportional to the button standoff:Zibut=k*soibut.[Ohm]  Equation (60)The proportionality provides a way to determine constant k after whichsomin can be obtained. With ∈0, Abut and ω known, ∈mud may bedetermined. The de-averaged standoff may be given by:daso=soibut(somin)−<soibut(somin)>,[m]  Equation (61)Equation (38) shows that daso can be approximated by the following (<x>denotes the average of x over the buttons):dasoibut=soibut(0)−<soibut(0)>.[m]  Equation (62)The de-averaged button impedance may be given by:daZibut=Zibut−<Zibut>.[Ohm]  Equation (63)The constant k can now be determined given that k may minimize the sumover the button electrodes of the squared difference between daZibut/kand dasoibut based on the following:k=arg min sumibut{(daZibut/k−dasoibut)²}[Ohm/m]  Equation (64)An algorithm, such as the Golden Section Search, can be used for theminimization.

With the thus-obtained k and Equation (41), the standoff of the buttonelectrodes may be determined. By fitting a quadratic and determining theminimum, somin may also be determined. In addition, ∈mud may bedetermined with the obtained value for k together with equations 39-41and with the help of known ∈0, Abut and ω.

FIG. 5 is a graph 500 depicting final so_(ibut) 578, quadratic curve580, and minimum 582. In this graph, standoff (y-axis) is plottedagainst button position (x-axis). In this example, the minimum 582 is at3.4, so_(min)=2.8 mm, and ∈_(mud)=10.4.

In the curvature mismatch technique, relative errors in the buttonimpedances may not propagate to so_(min), where the different buttonelectrodes have the same relative amplitude error. A relative error inthe button impedances may lead to a similar relative error in ∈_(mud).Outlier button impedances can be eliminated.

Other fitting functions, including smoothing and interpolationfunctions, may be used. Other means may be used to determine whichbutton electrode is closest to the formation, e.g. acoustic, mechanical.Other ways of matching Z_(ibut) and so_(ibut), such as fitting aquadratic therethrough and optimizing so_(min) until the x² terms match,may be used. Using a more complicated model between Z_(ibut) andso_(ibut) e.g., including second order terms and/or potentially based oncomputer simulations, such as finite elements, may be performed. Afterfinding so_(min), so_(min) _(_) _(initial)=so_(min) may be set andEquation 37 may be recalculated with the approximationso_(ibut)(so_(min))≈f_(ibut)(casingdiam, crvrad, so_(min) _(_)_(initial), xso_(min))+(so_(min)−so_(min) _(_) _(initial)). Othergeometries (button number, size, spacing, etc.) may be used. Thetechnique may be used, for example, in four terminal measurements withoil based mud (OBMI).

FIG. 6 is a flow chart depicting a method 600 of generating calibratedimages of a wellbore. The method 600 involves deploying (690) a downholetool into a cased portion of the wellbore (the downhole tool having atleast one sensor pad for measuring downhole parameters), obtaining (692)cased hole measurements in a cased hole portion of the wellbore with thesensor pad(s), and determining (694) cased hole parameters from thecased hole measurements. The method may also involve deploying (696) thedownhole tool into an open hole portion of the wellbore, obtaining (697)open hole measurements in an open hole portion of the wellbore with thesensor pad(s), determining (698) open hole parameters from the casedhole parameters, known parameters and the open hole measurements, andgenerating (699) downhole outputs from the determined open holeparameters. The method may be repeated or performed in any order.

Although only a few example embodiments have been described in detailabove, those skilled in the art will readily appreciate that manymodifications are possible in the example embodiments without materiallydeparting from this invention. Accordingly, the such modifications areintended to be included within the scope of this disclosure as definedin the following claims. In the claims, means-plus-function clauses areintended to cover the structures described herein as performing therecited function and not only structural equivalents, but alsoequivalent structures. Thus, although a nail and a screw may not bestructural equivalents in that a nail employs a cylindrical surface tosecure wooden parts together, whereas a screw employs a helical surface,in the environment of fastening wooden parts, a nail and a screw may beequivalent structures. It is the express intention of the applicant notto invoke 35 U.S.C. §112, paragraph 6 for any limitations of any of theclaims herein, except for those in which the claim expressly uses thewords ‘means for’ together with an associated function.

We claim:
 1. A method of generating calibrated downhole images of asubterranean formation (110) surrounding a wellbore (105), the methodcomprising: deploying (690) a downhole tool (132) into a cased holeportion (124) of the wellbore, the downhole tool having at least onesensor pad (117) for measuring downhole parameters, the at least onesensor pas comprising at least one button electrode and at least onereturn electrode; obtaining (692) cased hole measurements in the casedhole portion of the wellbore with the at least one sensor pad, the casedhole measurements comprising a cased hole impedance, wherein the casedhole impedance comprises one of a button impedance, a mud impedance andcombinations thereof; determining (694) cased hole parameters from thecased hole measurements; deploying (696) the downhole tool into an openhole portion of the wellbore; obtaining (697) open hole measurements inthe open hole portion (126) of the wellbore with the at least one sensorpad, the open hole measurements comprising an open hole impedance;determining (698) open hole parameters from the cased hole parameters,known parameters and the open hole measurements, the open holeparameters comprising a length Z90 of a vector orthogonal to a directionof a mud impedance vector, said orthogonal vector being obtained bydecomposing a button impedance vector in a complex plane; and generating(699) downhole outputs from the determined open hole parameters.
 2. Themethod of claim 1, further comprising determining formation resistivityfrom the vector length Z90.
 3. The method of claim 1, wherein each ofthe obtaining comprises passing a current from the at least one buttonelectrode to the at least one return electrode and measuring thecurrent.
 4. The method of claim 3, further comprising comparing thecurrent measured by a plurality of the at least one button electrode andat least one return electrode of the at least one sensor pad.
 5. Themethod of claim 1, further comprising determining a standoff betweeneach of the at least one button electrode and a measurement surface andadjusting for the standoff.
 6. The method of claim 1, further comprisinganalyzing one of the cased hole measurements, the cased hole parameters,the open hole measurements, the open hole parameters and the vectorlength Z90 and adjusting the one of the cased hole measurements, thecased hole parameters, the open hole measurements, the open holeparameters and vector length Z90.
 7. The method of claim 1, wherein ameasured button impedance equals a real button impedance.
 8. The methodof claim 1, wherein the cased hole impedance has an amplitude, amagnitude, a phase, and an angle.
 9. The method of claim 1, wherein theknown parameters comprise one of a casing curvature, a sensor padcurvature, a standoff, a mud angle and combinations thereof.
 10. Themethod of claim 9, further comprising determining a curvature mismatchbetween the casing curvature and the sensor pad curvature and adjustingthe cased hole parameters for the curvature mismatch.
 11. The method ofclaim 1, wherein the cased hole parameters comprise a mud angle, a mudpermittivity, a standoff, a gain factor, an amplitude offset andcombinations thereof.
 12. The method of claim 1, wherein the open holeparameters comprise an open hole amplitude, an open hole phase andcombinations thereof.
 13. A system for generating calibrated downholeimages of a subterranean formation (110) surrounding a wellbore (105),the system comprising: a downhole tool (132) positionable in a casedhole portion (124) and an open hole portion (126) of the wellbore (105),the downhole tool comprising: at least one sensor pad (117) formeasuring downhole parameters; at least one button electrode (140) onthe at least one sensor pad; at least one return electrode (136) on theat least one sensor pad; and electronics (142) in communication with theat least one button electrode and the at least one return electrode, theelectronics obtaining cased hole measurements in the cased hole portionof the wellbore and open hole measurements in the open hole portion ofthe wellbore with the at least one sensor pad, the cased holemeasurements comprising a cased hole impedance, wherein the cased holeimpedance comprises one of a button impedance, a mud impedance andcombinations thereof, the open hole measurements comprising an open holeimpedance, the electronics determining cased hole parameters from thecased hole measurements and open hole parameters from the cased holeparameters, known parameters and the open hole measurements, the openhole parameters comprising a length Z90 of a vector orthogonal to adirection of a mud impendance vector, said orthogonal vector beingobtained by decomposing a button impedance vector in a complex plane.14. The system of claim 13, further comprising at least one guardelectrode (138) between the at least one button electrode and the atleast one return electrode.
 15. The system of claim 13, furthercomprising at least one wear plate (130,131) extending from a front faceof the at least one sensor pad.
 16. The system of claim 13, furthercomprising an electrically insulated material (135) along a front face(134) of the at least one sensor pad, the at least one button electrodeand the at least one return electrode positionable in the insulatedmaterial.
 17. The system of claim 13, wherein a front face of the atleast one sensor pad has a curvature (φ_(p)), each of the at least onebutton electrode and the at least one return electrode positionablealong the curvature.
 18. The system of claim 13, wherein the at leastone sensor pad is positionable against a measurement surface (106, 111,115) via at least one leg (116).
 19. A method of generating calibrateddownhole images of a subterranean formation (110) surrounding a wellbore(105), the method comprising: deploying (690) a downhole tool (132) intoa cased hole portion (124) of the wellbore, the downhole tool having atleast one sensor pad (117) for measuring downhole parameters; obtaining(692) cased hole measurements in the cased hole portion of the wellborewith the at least one sensor pad, the cased hole measurements comprisinga cased hole impedance, wherein the cased hole impedance comprises oneof a button impedance, a mud impedance and combinations thereof;determining (694) cased hole parameters from the cased holemeasurements; deploying (696) the downhole tool into an open holeportion of the wellbore; obtaining (697) open hole measurements in theopen hole portion (126) of the wellbore with the at least one sensorpad, the open hole measurements comprising an open hole impedance;determining (698) open hole parameters from the cased hole parameters,known parameters and the open hole measurements, the open holeparameters comprising a length Z90 of a vector orthogonal to a directionof a mud impedance vector, said orthogonal vector being obtained bydecomposing a button impedance vector in a complex plane; generating(699) downhole outputs from the determined open hole parameters;determining formation resistivity from the vector length Z90; andanalyzing one of the cased hole measurements, the cased hole parameters,the open hole measurements, the open hole parameters and the vectorlength Z90 and adjusting the one of the cased hole measurements, thecased hole parameters, the open hole measurements, the open holeparameters and the vector length Z90; wherein the cased hole impedancehas an amplitude, a magnitude, a phase, and an angle; wherein the knownparameters comprise one of a casing curvature, a sensor pad curvature, astandoff, a mud angle and combinations thereof; wherein the cased holeparameters comprise a mud angle, a mud permittivity, a standoff, a gainfactor, an amplitude offset and combinations thereof; and wherein theopen hole parameters comprise an open hole amplitude, an open hole phaseand combinations thereof.